A coupled ring-core fiber with extremely negative dispersion is proposed, simulated, and analyzed. Featured by two coaxial Ge-doped silica rings, this fiber can support highly dispersive orbital angular momentum (OAM) modes. The dependence of dispersion characteristics on the variable parameters of the fiber is theoretically and systematically investigated, including the geometry parameters and the mole fraction of GeO2. The order of the supported OAM modes in this fiber can be up to 100, achieving a highly negative dispersion reaching to -44,579 ps/(nm·km) at 1348.4 nm. The operation wavelength and the value of dispersion can be adjusted according to the need by selecting the fiber parameters. Furthermore, by adjusting the GeO2 concentration, the operation window of the OAM65,1 mode can be tuned in a much wider range from 1212.4 nm to 1666.2 nm.
Research Article
Germanium-Doped Coupled Ring-Core Fiber for Extremely Dispersive Orbital Angular Momentum Modes
https://doi.org/10.21203/rs.3.rs-1341308/v1
This work is licensed under a CC BY 4.0 License
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posted 24 Feb, 2022
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With the exponentially increased demand on information transmission capacity, space division multiplexing (SDM) has a promising potential to remove the obstacles in the current optical communication systems 1–5. Orbital angular momentum (OAM) modes propagating with the unique helical phase profiles have been applied as a multiplexing technique in the spatial domain. The OAM mode with certain topological charge l generates by linearly combining even and odd modes of the fiber eigenmodes (EH and HE modes) carrying the same number of intensity rings of the mode m as follow:
where the ‘±’ in superscript of ‘OAM’ is the direction of the circular polarization, the subscript ‘±’ before l denotes the rotation direction of wavefront. OAM modes with varied l are mutually orthogonal, contributing to their multiplexing along the same beam axis and demultiplexing with low crosstalk, which introduces a new degree of freedom for SDM systems 6–8. With the helical phase form, the modal distribution of the OAM mode has a donut shape and a central phase singularity. After decades of profound research, the promising applications of OAM modes have been extended tremendously to classical and quantum communications 9–11, optical trapping 12, 13, and sensing 14, 15.
Ring-core fiber with the annular index distribution has been demonstrated to propagate OAM modes stably, which introduces a feasible medium for OAM propagation 16–18. S. Ramachandran, et al. experimentally demonstrated that ring-core fiber can generate and propagate OAM modes over long distance with unprecedented stability in 2009 19. Recent report has shown that > 1000 information-carrying OAM states can be supported in an arsenic trisulfide ring fiber with air fiber core 20, and highly coherent two-octave-spanning OAM spectral broadening could be formed 21, 22. Silica (SiO2) has been widely used in manufacturing optical fiber by the industry and research for decades due to its favorable properties and mature preparation methods. Ge-doped silica, which is well compatible with the common fiber material silica, is also a popular substance in fiber research and drawing 23–27. In addition, due to the adjustable mole fraction of GeO2, the material refractive index of it can be tuned flexibly.
Chromatic dispersion (CD) is one of the important design issues for optical fiber in various applications, such as ultrafast optical oscilloscope 28, optical correlation 29, true time-delay beam former 30, and phased array antenna systems 31. To tailor the dispersion, which is usually positive in the optical fiber communication systems, fiber with large negative dispersion needs to be placed in serial along the optical fiber link. In the recent research and development, fibers with large negative dispersion for the fundamental fiber eigenmode have been reported 32–36.
In this paper, a highly dispersive coupled ring-core fiber (HD-CRF) based on silica and germanium-doped (Ge-doped) silica is proposed. This fiber with two coupled high-index rings can support high order OAM modes characterized by extremely negative dispersion. With properly optimized fiber parameters, the dispersion of OAM100,1 mode reaches to -44,579 ps/(nm·km) at the wavelength 1348.4 nm. The feasibility of the proposed fiber shows its potential dispersion-adjusting applications in a variety of OAM transmission systems based on optical fiber.
Structure of the designed OAM HD-CRF
The schematic structure and material refractive index distribution design of the OAM HD-CRF at 1550 nm is depicted in Fig. 1. Here, r1, r2, r3, r4, Δr1, Δr2 and d are variable parameters and the fiber cladding radius r5 is set as a constant 62.5 µm. This constant is matched to the radius of the common single-mode optical fiber (SMF). The proposed HD-CRF consists of two high-index pure GeO2 rings in the background of undoped silica. The material refractive indices of the high-index rings and the low-index cladding show a step, which influences the order of the OAM mode supported in this fiber and can be adjusted by changing the concentration of GeO2 doped in silica.
Property of supported OAM modes
Figure 2(a) depicts the process of negative chromatic dispersion generated in the proposed OAM fiber and Fig. 2(b) shows the simulated mode field intensity and phase distributions of OAM modes with topological charge |l| = 25, 50, 75, 100 supported in this fiber at the most dispersive wavelength by the full vector finite element method (FEM). Here, the modal intensity has an annular distribution, while the phase front is in a helical form. Moreover, when l1 > l2, the radii of two high-index rings supporting OAM modes with the topological charge l1 are larger than those supporting OAM modes with l2. Table 1 summarizes the optimized geometry parameters of HD-CRFs supporting the above OAM modes.
Figure 3(a) illustrates the effective refractive indices neff with intensity distributions versus wavelength for the OAM100,1 mode supported in the inner ring, outer ring and the proposed fiber with two rings by implementing Sellmeier equations 37, 38. As wavelength increases, the neff of the OAM mode in the ring near the fiber core reduces relatively drastically, while the neff of the same OAM mode in the high-index ring near the cladding decreases more gradual. Therefore, by optimizing the fiber parameters, the neff curves of OAM100,1 mode supported in two rings respectively can intersect at a desired wavelength (λ0). Consequently, at this wavelength λ0, there is a great slope variation in the composite index curve of the OAM100,1 mode in the designed dual-ring fiber.
|
Mode |
OAM25,1 |
OAM50,1 |
OAM75,1 |
OAM100,1 |
|---|---|---|---|---|
|
r1 (µm) |
11.5 |
23.5 |
35 |
49 |
|
r2 (µm) |
13 |
25 |
36.5 |
50.5 |
|
r3 (µm) |
16.5 |
28.5 |
40 |
54 |
|
r4 (µm) |
17.3 |
29.5 |
41.1 |
55.2 |
|
λ0 (nm) |
1469.6 |
1402.8 |
1360.2 |
1348.4 |
|
min. dispersion (ps/(nm·km)) |
-113,619 |
-90,512 |
-76,282 |
-44,579 |
Figure 3(b) depicts the effective indices neff of eigenmodes near the HE101,1 mode, which compose the OAM100,1 mode in this fiber. The neff difference between eigenmodes (EH98,1 mainly supported in the inner ring, HE101,1, and EH96,1 mainly supported in the outer ring) for HD-CRF with the same structure parameters can be maintained above ~ 10− 4 over the operating wavelength range to ensure the good separation between eigenmodes.
Figure 4 compares the properties of different order OAM modes supported in the proposed HD-CRF with geometry parameters summarized in Table 1, including the effective mode area Aeff, ratio of power integral in the high-index ring near the core to that in ring near the fiber cladding, and chromatic dispersion D. Here, the OAM modes dispersion 39 are calculated from the effective refractive index neff with wavelength as independent variable as follows:
where λ represents the wavelength, c denotes the velocity of the light propagating in the vacuum.
For the sake of clarity, the brief description of the principle for the proposed fiber is explained as follows. As shown in Fig. 3(a) and Fig. 4, when λ < λ0, the mode supported in this fiber is basically confined in the ring near the fiber core, which leads to the power ratio relatively high. Near the crossing wavelength λ0, a significant optical coupling occurs. When the mode field distribution moves to ring near the cladding, Aeff rapidly increases, and the power ratio decreases to 1. The dispersion of this OAM mode thus has an extremely negative value at the composite refractive index abrupt wavelength λ0 as shown in Fig. 4(c). As the wavelength further increases, the mode is mainly guided in the ring near the cladding. With the optimized geometrical parameters, the D of the OAM100,1 mode reaches downward to -44,579 ps/(nm·km) at 1348.4 nm. Table 1 lists the minimum dispersions and their corresponding wavelength λ0 of the other OAM modes (l = 25, 50, 75).
Dispersion characteristic
As discussed above, by varying the neff profiles of the two rings, the value and the position of the minimum dispersion could be changed accordingly. We further investigate the influence of the fiber geometrical and material parameters on chromatic dispersion for the OAM100,1 mode.
Figure 5 illustrates the dependence of the chromatic dispersion on the thickness of two concentric germanium-doped rings (Δr1 and Δr2). When Δr1 and Δr2 increase simultaneously, the most dispersive wavelength is shifted to short wavelength with more negative CD. The minimum dispersion of the OAM100,1 mode is reduced dramatically from − 14,145 to -44,579 ps/(nm·km), and λ0 moves from 1629.2 nm to 1348.4 nm. However, larger width has stronger restrictions on the modes which further limit the mode transition between the inner ring and the outer ring.
Figure 6 displays the dispersion profiles of the OAM100,1 mode with different distances between the two Ge-doped rings d from 4.6 to 5 µm while fixing the inner ring and width (r1 = 49 µm, r2 = 50.5 µm and Δr2 = 1.2 µm). When the outer high-index ring is moved away from the fiber core, the position of λ0 blue-shifts from 1436.2 nm to 1348.4 nm and the most negative dispersion varies from − 9,255 to -44,579 ps/(nm·km). Furthermore, the sacrifice between the full width at half maximum (FWHM) of the dispersive window and the extreme value of CD need to be taken into consideration in the optimization process 40. When the dispersion is reduced from − 9,255 to -44,579 ps/(nm·km), the FWHM of the dispersive window reduces from 3.4 to 0.6 nm.
Figure 7 depicts the CD profile of the OAM100,1 mode in the designed HD-CRF with two doped ring moving towards the cladding in a step of 0.5 µm, while the relative location and thickness of the two regions are constant (d = 5 µm, Δr1 = 1.5 µm, Δr2 = 1.2 µm). With the two Ge-doped rings moving 1 µm in the radial direction, λ0 is increased by 65.6 nm and the extremum of the dispersion rises from − 44,579 to -28,895 ps/(nm·km). Here, the effective refractive index curves for the OAM100,1 mode supported independently in the inner and outer rings simultaneously move up, and the decrease in their refractive indices with wavelength is relatively slower. However, the slope of neff in the inner heavily doped ring increases more. As the slope difference between the two rings decreases, the most dispersive wavelength shifts and the extreme of CD increases.
Due to the variable mole fraction of GeO2, the dispersion characteristics of the OAM modes supported in the proposed heavily germanium-doped HD-CRF can be further tailored. To prevent the appearance of radially higher order modes (m ≥ 2), the thickness of the ring with high index is carefully controlled, which guarantees that only radial fundamental (m = 1) OAM modes are supported. As the mole fraction of GeO2 decreases, the refractive index difference between the ring and the cladding decreases, and thus the OAM modes supported in the low-doped ring fiber are less than that in the high-doped ring fiber under certain ring width. The chromatic dispersion curves of the OAM65,1 mode under varying concentration of GeO2 in the two high-index rings are depicted in Fig. 8. As the mole fraction of GeO2 increases from 50–100%, the dispersion window is shifted by 453.8 nm to the longer wavelength region and achieves a larger negative dispersion, which is from − 10,773 to -20,152 ps/(nm·km). The structure parameters of the fiber with higher mole fraction can be further optimized to acquire larger negative dispersion. However, there is an increment in the loss as the mole fraction increases, which cannot be neglected in fiber parameters selection.
In summary, we investigated a dual-ring HD-CRF with germanium-doped silica as high-index materials, which could support highly dispersive OAM modes. With the intense coupling between the modes distributed in the two rings, the negative dispersion for the OAM100,1 mode reaches to -44,579 ps/(nm·km) at 1348.4 nm. The numerical results indicate that the working wavelength window is mainly dependent on the effective refractive indices of the OAM modes in the two rings. Therefore, the tunable range of the most dispersive wavelength can be greatly enlarged by changing the doping concentration of the GeO2 in the two rings. The highly dispersive OAM fiber can be potentially applied in plenty of fields, such as the dispersion compensation and optical buffering 41.
The HD-CRF simulations were carried out using full vector finite element method (FEM). First, the cross-section of the fiber is divided into different homogeneous subspaces, so that a simple finite element mesh is obtained after distribution. In order to describe the strength and distribution of the magnetic or electric field more precisely, as the distance of the subspace to the center is closer, their size will gradually decrease. Then use Maxwell's equation to discretize each subspace element to get a set of fundamental matrices. The combination of these fundamental matrices creates a global matrix system for the entire structure. Finally, the effective refractive index and distribution of the amplitudes and polarizations of the modes are numerically calculated, taking into account the continuity conditions at each subspace boundary.
Data availability
All data generated or analysed during this study are included in this published article.
Author contributions statement
All the authors contributed equally in this work.
Competing interests
The authors declare no competing interests.
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No competing interests reported.